Info and Procedure (Method)
Full table of instructions
Means plots
Hypotheses
Results
- The first set of models
- The second set of models
Discussion

dat_full <- preprocessing(root_dir, data_dir)
dat <- subset(dat_full, subset=RT>0 & RT<25000) # loses 2 RTs that were judged to be outliers 25978 49871
dat_borderline <- dat
dat <- perSubjectTrim.fnc(dat, response='RT', subject='Subject', trim = 2.5)$data

## ......
## n.removed = 204 
## percent.removed = 2.797203

dat$RT_log <- log(dat$RT)

Info and Procedure (Method)

The main aim of this experiment was to see whether vague instructions would confer advantages over crisp alternatives when all instructions used numerals, and when there were vague and crisp versions of each of the comparison and matching strategies.

On each trial, first the referring expression that constituted the instruction for that trial was displayed (e.g., “Choose a square with about 10 dots”). Participants then pressed a key to indicate that they had read the instruction.

The instruction remained on screen, and after 1000 ms, the arrays appeared (see Figure below).

Response time was measured from the presentation of the arrays until the keypress indicating the participant’s choice. The trial would timeout after 60 seconds if there was no response.

In this experiment, no feedback was given. This was because, in the vague conditions, we did not regard any response as “correct” or “incorrect”, but instead as “expected response”; “borderline response”; and “extreme response”, and we did not want to draw participants’ attention to this distinction explicitly. Which choice the participant made was recorded for analysis.

Full table of instructions

The instructions for this experiment were similar to C_exp_2, except that we changed “fewer than” to “far fewer than”: this was because “far fewer than” can have borderline cases whereas “fewer than” cannot have borderline cases.

Full table of instructions
Item	Quantity	Selection	Crisp	Vague
06:15:24	Small	Comparison	Choose a square with fewer than 20 dots	Choose a square with far fewer than 20 dots
06:15:24	Small	Matching	Choose a square with 6 dots	Choose a square with about 10 dots
06:15:24	Large	Comparison	Choose a square with more than 10 dots	Choose a square with far more than 10 dots
06:15:24	Large	Matching	Choose a square with 24 dots	Choose a square with about 20 dots
16:25:34	Small	Comparison	Choose a square with fewer than 30 dots	Choose a square with far fewer than 30 dots
16:25:34	Small	Matching	Choose a square with 16 dots	Choose a square with about 20 dots
16:25:34	Large	Comparison	Choose a square with more than 20 dots	Choose a square with far more than 20 dots
16:25:34	Large	Matching	Choose a square with 34 dots	Choose a square with about 30 dots
26:35:44	Small	Comparison	Choose a square with fewer than 40 dots	Choose a square with far fewer than 40 dots
26:35:44	Small	Matching	Choose a square with 26 dots	Choose a square with about 30 dots
26:35:44	Large	Comparison	Choose a square with more than 30 dots	Choose a square with far more than 30 dots
26:35:44	Large	Matching	Choose a square with 44 dots	Choose a square with about 40 dots
36:45:54	Small	Comparison	Choose a square with fewer than 50 dots	Choose a square with far fewer than 50 dots
36:45:54	Small	Matching	Choose a square with 36 dots	Choose a square with about 40 dots
36:45:54	Large	Comparison	Choose a square with more than 40 dots	Choose a square with far more than 40 dots
36:45:54	Large	Matching	Choose a square with 54 dots	Choose a square with about 50 dots

Means plots

dat_plot <- summarySEwithin2(dat, measurevar="RT_log", withinvars=c("Vagueness", "Selection", "Item"), idvar="Subject")

dat$Condition <- as.factor(paste(dat$Item, dat$Vagueness))
ggplot(data=dat, mapping=aes(x=Item, y=RT_log, group=Vagueness)) +
  theme_bw() +  theme(panel.grid.major = element_blank(), panel.grid.minor = element_blank(), aspect.ratio = 1) +
  facet_wrap(~Selection) +
  stat_summary(mapping=aes(y=RT_log, x=Item, group=Vagueness, colour=Vagueness), fun.y=mean, geom="line", lwd=2) +
  geom_smooth(mapping=aes(y=RT_log, x=Item, group=Vagueness, colour=Vagueness), method="loess")

## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : pseudoinverse used at 0.985

## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : neighborhood radius 2.015

## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : reciprocal condition number 4.5804e-15

## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : There are other near singularities as well. 4.0602

## Warning in predLoess(object$y, object$x, newx = if
## (is.null(newdata)) object$x else if (is.data.frame(newdata))
## as.matrix(model.frame(delete.response(terms(object)), : pseudoinverse used
## at 0.985

## Warning in predLoess(object$y, object$x, newx = if
## (is.null(newdata)) object$x else if (is.data.frame(newdata))
## as.matrix(model.frame(delete.response(terms(object)), : neighborhood radius
## 2.015

## Warning in predLoess(object$y, object$x, newx = if
## (is.null(newdata)) object$x else if (is.data.frame(newdata))
## as.matrix(model.frame(delete.response(terms(object)), : reciprocal
## condition number 4.5804e-15

## Warning in predLoess(object$y, object$x, newx = if
## (is.null(newdata)) object$x else if (is.data.frame(newdata))
## as.matrix(model.frame(delete.response(terms(object)), : There are other
## near singularities as well. 4.0602

## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : pseudoinverse used at 4.015

## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : neighborhood radius 2.015

## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : reciprocal condition number 4.4845e-15

## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : There are other near singularities as well. 1

## Warning in predLoess(object$y, object$x, newx = if
## (is.null(newdata)) object$x else if (is.data.frame(newdata))
## as.matrix(model.frame(delete.response(terms(object)), : pseudoinverse used
## at 4.015

## Warning in predLoess(object$y, object$x, newx = if
## (is.null(newdata)) object$x else if (is.data.frame(newdata))
## as.matrix(model.frame(delete.response(terms(object)), : neighborhood radius
## 2.015

## Warning in predLoess(object$y, object$x, newx = if
## (is.null(newdata)) object$x else if (is.data.frame(newdata))
## as.matrix(model.frame(delete.response(terms(object)), : reciprocal
## condition number 4.4845e-15

## Warning in predLoess(object$y, object$x, newx = if
## (is.null(newdata)) object$x else if (is.data.frame(newdata))
## as.matrix(model.frame(delete.response(terms(object)), : There are other
## near singularities as well. 1

## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : pseudoinverse used at 0.985

## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : neighborhood radius 2.015

## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : reciprocal condition number 2.0732e-15

## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : There are other near singularities as well. 1

## Warning in predLoess(object$y, object$x, newx = if
## (is.null(newdata)) object$x else if (is.data.frame(newdata))
## as.matrix(model.frame(delete.response(terms(object)), : pseudoinverse used
## at 0.985

## Warning in predLoess(object$y, object$x, newx = if
## (is.null(newdata)) object$x else if (is.data.frame(newdata))
## as.matrix(model.frame(delete.response(terms(object)), : neighborhood radius
## 2.015

## Warning in predLoess(object$y, object$x, newx = if
## (is.null(newdata)) object$x else if (is.data.frame(newdata))
## as.matrix(model.frame(delete.response(terms(object)), : reciprocal
## condition number 2.0732e-15

## Warning in predLoess(object$y, object$x, newx = if
## (is.null(newdata)) object$x else if (is.data.frame(newdata))
## as.matrix(model.frame(delete.response(terms(object)), : There are other
## near singularities as well. 1

## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : pseudoinverse used at 4.015

## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : neighborhood radius 2.015

## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : reciprocal condition number 3.4466e-15

## Warning in simpleLoess(y, x, w, span, degree = degree, parametric =
## parametric, : There are other near singularities as well. 1

## Warning in predLoess(object$y, object$x, newx = if
## (is.null(newdata)) object$x else if (is.data.frame(newdata))
## as.matrix(model.frame(delete.response(terms(object)), : pseudoinverse used
## at 4.015

## Warning in predLoess(object$y, object$x, newx = if
## (is.null(newdata)) object$x else if (is.data.frame(newdata))
## as.matrix(model.frame(delete.response(terms(object)), : neighborhood radius
## 2.015

## Warning in predLoess(object$y, object$x, newx = if
## (is.null(newdata)) object$x else if (is.data.frame(newdata))
## as.matrix(model.frame(delete.response(terms(object)), : reciprocal
## condition number 3.4466e-15

## Warning in predLoess(object$y, object$x, newx = if
## (is.null(newdata)) object$x else if (is.data.frame(newdata))
## as.matrix(model.frame(delete.response(terms(object)), : There are other
## near singularities as well. 1

Hypotheses

We formulated the following hypotheses:

[Hypothesis 1] Crisp/Vague RT:
- Vague instructions should result in faster responses than crisp instructions.
[Hypothesis 2] Comparison/Matching RT:
- Instructions that allow comparison should result in faster responses than instructions that necessitate matching.
[Hypothesis 3] Vagueness effect differs between comparison and matching:
- The effects of vagueness should differ when the selection task is comparison versus when it is matching.

Results

Response times were trimmed at 25,000 ms, resulting in the loss of 2 trials that were judged to have outlier RTs. The remaining data were trimmed separately for each participant at 2.5 standard deviations below and above that participant’s mean RT, leading to the loss of 204 trials (2.8% of the remaining trials). RTs in milliseconds were log-transformed so that they better approximated a normal distribution for the purposes of analysis.

The first set of models

dat_model <- dat
dat_model$c_Vag <- ifelse(dat_model$Vagueness=="Crisp", -0.5, 0.5)
dat_model$c_Sel <- ifelse(dat_model$Selection=="Comparison", -0.5, 0.5)
dat_model$c_Itm <- ifelse(dat_model$Item=="06:15:24", -.75, ifelse(dat_model$Item=="16:25:34", -.25, ifelse(dat_model$Item=="26:35:44", .25, .75)))
rtlmer = lmerTest::lmer(data=dat_model, RT_log ~ c_Vag * c_Sel + c_Itm + (1 + c_Vag * c_Sel + c_Itm | Subject))
pretty_coef_table(rtlmer, "rt_lmer_full_model_all_items")

rt_lmer_full_model_all_items
term	𝛽	s.e.	d.f.	𝑡	𝑝	Pr(>\|t\|)	sig.
c_Vag	-0.0051	0.014	36.94	-0.37	7.16e-01	0.716
c_Sel	0.1619	0.026	36.99	6.24	2.97e-07	<0.001	＊＊＊
c_Itm	0.0580	0.018	36.92	3.23	2.60e-03	<0.01	＊＊
c_Vag:c_Sel	0.1324	0.031	36.87	4.23	1.49e-04	<0.001	＊＊＊

dat_model <- subset(dat, subset=Selection=='Comparison')
dat_model$c_Vag <- ifelse(dat_model$Vagueness=="Crisp", -0.5, 0.5)
dat_model$c_Itm <- ifelse(dat_model$Item=="06:15:24", -.75, ifelse(dat_model$Item=="16:25:34", -.25, ifelse(dat_model$Item=="26:35:44", .25, .75)))
comp_lmer <- lmerTest::lmer(data=dat_model, RT_log ~ c_Vag + c_Itm + (1 + c_Vag + c_Itm | Subject))
pretty_coef_table(comp_lmer, "rt_lmer_for_comparison_all_items")

rt_lmer_for_comparison_all_items
term	𝛽	s.e.	d.f.	𝑡	𝑝	Pr(>\|t\|)	sig.
c_Vag	-0.071	0.020	36.99	-3.52	1.15e-03	<0.01	＊＊
c_Itm	0.090	0.020	37.02	4.61	4.68e-05	<0.001	＊＊＊

dat_model <- subset(dat, subset=Selection=='Matching')
dat_model$c_Vag <- ifelse(dat_model$Vagueness=="Crisp", -0.5, 0.5)
dat_model$c_Itm <- ifelse(dat_model$Item=="06:15:24", -.75, ifelse(dat_model$Item=="16:25:34", -.25, ifelse(dat_model$Item=="26:35:44", .25, .75)))
match_lmer <- lmerTest::lmer(data=dat_model, RT_log ~ c_Vag + c_Itm + (1 + c_Vag + c_Itm | Subject))
pretty_coef_table(match_lmer, "rt_lmer_for_matching_all_items")

rt_lmer_for_matching_all_items
term	𝛽	s.e.	d.f.	𝑡	𝑝	Pr(>\|t\|)	sig.
c_Vag	0.062	0.021	36.32	2.91	6.14e-03	<0.01	＊＊
c_Itm	0.025	0.022	36.62	1.12	2.71e-01	0.271

Considering the first set of models:

[Hypothesis 1] Crisp/Vague RT:
- Vague instructions should result in faster responses than crisp instructions : Vague instructions resulted in faster responses than crisp instructions but the difference was not significant (p=0.716).
[Hypothesis 2] Comparison/Matching RT:
- Instructions that allow comparison should result in faster responses than instructions that necessitate matching : comparison instructions resulted in faster responses than matching instructions, and the difference was significant (p<0.001).
[Hypothesis 3] Vagueness effect differs between comparison and matching
- The effects of vagueness should differ when the selection task is comparison versus when it is matching : Vagueness exerted effects in different directions for the two selection tasks (p<0.001). Separate analyses at each level of selection provided evidence that vague instructions resulted in faster responses than crisp instructions for the comparison condition (p<0.01); and slower responses than crisp instructions in the matching condition (p<0.01).

The second set of models

The second set of models considered were conducted to test for main effects in the presence of interactions involving those main effects (after Levy, 2018).

Testing for a main effect of X when X is a two-level factor and the model also includes a main effect of a three-level factor and the interaction between X and Y.

Testing for a main effect of Vagueness when Vagueness is a two-level factor and the model also includes a main effect of a four-level factor Item and the interaction between Vagueness and Item.

Item.numeric <- sapply(dat$Item, function(i) contr.sum(4)[i,])
dat$Item1 <- Item.numeric[1,]
dat$Item2 <- Item.numeric[2,]
dat$Item3 <- Item.numeric[3,]
m0 <- lm(RT_log ~ 
           Item1 + Vagueness:Item1 + 
           Item2 + Vagueness:Item2 + 
           Item3 + Vagueness:Item3 +
           Selection +
           Selection:Vagueness, 
         dat)
contrasts(dat$Item) <- contr.sum(4)
m.test <- lm(RT_log ~ 
               Item +
           Vagueness:Item1 + 
           Vagueness:Item2 + 
           Vagueness:Item3 +
           Selection +
           Selection:Vagueness, 
         dat)
m1 <- lm(RT_log ~ 
           Item1 + Vagueness:Item1 + 
           Item2 + Vagueness:Item2 + 
           Item3 + Vagueness:Item3 +
           Selection +
           Selection:Vagueness +
           Vagueness,
         dat)
m100 <- lm(RT_log ~
             Item +
             Vagueness +
             Item:Vagueness+
             Selection+
             Selection:Vagueness,
           dat)
#m1 and m100 are the same model
anova(m0,m1,test="F")

## Analysis of Variance Table
## 
## Model 1: RT_log ~ Item1 + Vagueness:Item1 + Item2 + Vagueness:Item2 + 
##     Item3 + Vagueness:Item3 + Selection + Selection:Vagueness
## Model 2: RT_log ~ Item1 + Vagueness:Item1 + Item2 + Vagueness:Item2 + 
##     Item3 + Vagueness:Item3 + Selection + Selection:Vagueness + 
##     Vagueness
##   Res.Df    RSS Df Sum of Sq      F   Pr(>F)    
## 1   7080 2694.5                                 
## 2   7079 2689.7  1    4.8392 12.736 0.000361 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Item.numeric <- sapply(dat$Item, function(i) contr.sum(4)[i,])
dat$Item1 <- Item.numeric[1,]
dat$Item2 <- Item.numeric[2,]
dat$Item3 <- Item.numeric[3,]
m3 <- lm(RT_log ~ 
           Item1 + 
           Item2 +  
           Item3 , 
         dat)
contrasts(dat$Item) <- contr.sum(4)
m4 <- lm(RT_log ~
           Item,
         dat)
# m3 and m4 are the same model

Item.numeric <- sapply(dat$Item, function(i) contr.sum(4)[i,])
dat$Item1 <- Item.numeric[1,]
dat$Item2 <- Item.numeric[2,]
dat$Item3 <- Item.numeric[3,]

m.full <- lm(RT_log ~
             Item +
             Vagueness +
             Selection + 
             Item:Vagueness +
             Item:Selection + 
             Vagueness:Selection,
           dat)
m.reduced <- lm(RT_log ~
                Item1 + 
                Item2 + 
                Item3 + 
                Vagueness:Item1 + 
                Vagueness:Item2 +
                Vagueness:Item3 +
                Vagueness:Selection +
                Selection +
                Item:Selection
                ,dat  )
anova(m.reduced, m.full, test="F")

## Analysis of Variance Table
## 
## Model 1: RT_log ~ Item1 + Item2 + Item3 + Vagueness:Item1 + Vagueness:Item2 + 
##     Vagueness:Item3 + Vagueness:Selection + Selection + Item:Selection
## Model 2: RT_log ~ Item + Vagueness + Selection + Item:Vagueness + Item:Selection + 
##     Vagueness:Selection
##   Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
## 1   7077 2686.7                                  
## 2   7076 2681.8  1    4.8767 12.867 0.0003367 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

D_exp_2

Matt Green